Standard +0.8 This question requires recognizing that the numerator is cos(2x) and denominator is 1+sin(2x), then using substitution u=1+sin(2x) to integrate. While the double angle identities are standard C4 content, spotting the structure and executing the substitution with correct limits requires solid technique and algebraic fluency beyond routine exercises.
\(\cos2x=1-2\sin^2x\) or \((1+)\sin2x=(1+)2\sin x\cos x\) seen; if B0B0M0A0, SC4 for \(F[x]=\frac{1}{2}\ln(1+2\sin x\cos x)\) or \(\frac{1}{2}\ln(1+\sin2x)\); final mark may still be awarded
B1*
Numerator and denominator both correct in the integral soi
\(F[x]=k\ln(1+\sin2x)\) soi
M1dep*
or \(k\ln(1+u)\) or \(k\ln(u)\) following their substitution www
Correct use of limits www; minimum working: \(\frac{1}{2}\ln2-\frac{1}{2}\ln1\) or \(\frac{1}{2}\ln(1+1)\) oe
[5]
# Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int\frac{\cos2x}{1+\sin2x}(dx)$ | B1* | $\cos2x=1-2\sin^2x$ or $(1+)\sin2x=(1+)2\sin x\cos x$ seen; if B0B0M0A0, SC4 for $F[x]=\frac{1}{2}\ln(1+2\sin x\cos x)$ or $\frac{1}{2}\ln(1+\sin2x)$; final mark may still be awarded |
| | B1* | Numerator and denominator both correct in the integral soi |
| $F[x]=k\ln(1+\sin2x)$ soi | M1dep* | or $k\ln(1+u)$ or $k\ln(u)$ following their substitution www |
| $k=\frac{1}{2}$ | A1 | Correct $k$ for their substitution |
| $\frac{1}{2}\ln(1+\sin(\frac{\pi}{2}))-\frac{1}{2}\ln(1+0)=\frac{1}{2}\ln2$ | A1 AG | Correct use of limits www; minimum working: $\frac{1}{2}\ln2-\frac{1}{2}\ln1$ or $\frac{1}{2}\ln(1+1)$ oe |
| **[5]** | | |